Quantum error mitigation using energy sampling and extrapolation enhanced Clifford data regression

TL;DR

The paper tackles the challenge of quantum error mitigation for quantum chemistry on NISQ devices by extending Clifford Data Regression (CDR) with two strategies: Energy Sampling (ES) and Non-Clifford Extrapolation (NCE). It implements these on the H$_4$ system with a tiled Unitary Product State (tUPS) Ansatz under an IBM Torino noise model, benchmarking how training-set design and circuit depth affect mitigation. The results show that ES consistently improves performance at low quantum cost, while NCE can yield even higher accuracy at greater computational expense, with mixed results across circuit depths. Together, these enhancements demonstrate a path toward more reliable VQE-based chemistry computations on near-term quantum hardware and motivate future work combining ES and NCE with more advanced regression models.

Abstract

Error mitigation is essential for the practical implementation of quantum algorithms on noisy intermediate-scale quantum (NISQ) devices. This work explores and extends Clifford Data Regression (CDR) to mitigate noise in quantum chemistry simulations using the Variational Quantum Eigensolver (VQE). Using the H$_4$ molecule with the tiled Unitary Product State (tUPS) ansatz, we perform noisy simulations with the ibm torino noise model to investigate in detail the effect of various hyperparameters in CDR on the error mitigation quality. Building on these insights, two improvements to the CDR framework are proposed. The first, Energy Sampling (ES), improves performance by selecting only the lowest-energy training circuits for regression, thereby further biasing the sample energies toward the target state. The second, Non-Clifford Extrapolation (NCE), enhances the regression model by including the number of non-Clifford parameters as an additional input, enabling the model to learn how the noisy-ideal mapping evolves as the circuit approaches the optimal one. Our numerical results demonstrate that both strategies outperform the original CDR.

Figures (8)

• Figure 1: (a) Traditional CDR method: A set of (near-)Clifford circuits are executed on both a quantum device and a classical computer, and the resulting noisy and noise-free expectation values are used for regression. (b) Energy Sampling CDR: Noise-free expectation values from $M$ classically simulated (near-)Clifford circuits are obtained, and the $N<M$ lowest ones with their noisy counterparts are used for regression. (c) Non-Clifford Extrapolation CDR: A regression model incorporates the number of non-Clifford parameters $k$ as an additional feature in the training set, enabling the model to learn the dependence between noise-free and noisy expectation values in the low-$k$ regime and extrapolate to the target case $k=n$.
• Figure 2: Energy differences of noise-free and CDR-mitigated ground state energies for H$_4(4,4)$ with a tUPS Ansatz of layers $L = 2$ and $3$. The dependence of the mitigation with respect to the training set size $N$ is shown. For each system a linear (solid lines) and quadratic (dashed lines) model was used as well as with (blue lines) and without (red lines) biasing. The number of non-Clifford parameters was set to $k=4$ and $k=6$ for $L=2$ and $L=3$, respectively.
• Figure 3: Energy differences between noise-free and mitigated ground state energies for H$_4$ plotted as a function of the number of non-Clifford parameters $k$, using both 2-layer and 3-layer tUPS Ansätze. For each system, both linear (solid lines) and quadratic (dashed lines) regression models were performed, with and without biasing (blue and red lines, respectively). CDR was performed using $N = 153$ samples in the training datasets for both 2- and 3-layer tUPS ansatze.
• Figure 4: Absolute energy errors of the CDR-mitigated results with both 2-layer and 3-layer tUPS Ansätze, plotted as a function of training set size $N$. CDR was implemented with (ES, blue lines) and without (traditional, red lines) our energy sampling strategy. All training circuits were generated using biased preparation, with both linear (solid lines) and quadratic (dashed lines) models employed. For the ES-CDR, $N$ lowest-energy circuits were selected from a biased data pool of $M = 153$ samples. For $L=2$ and $L=3$, the number of the non-Clifford parameters $k$ is fixed at $4$ and $6$, respectively.
• Figure 5: Absolute energy errors for a fixed number of selected samples $N$, with the data pool size $M$ varying from $N$ to $1000$. For both 2-layer and 3-layer tUPS, $N$ is fixed at 30 (blue), 50 (green), and 80 (red). For 2-layer and 3-layer cases, the number of the non-Clifford parameters $k$ are fixed at $4$ and $6$, respectively.